Frequent pattern mining

ABSTRACT

A system for frequent pattern mining uses two layers of processing: a plurality of computing nodes, and a plurality of processors within each computing node. Within each computing node, the data set against which the frequent pattern mining is to be performed is stored in shared memory, accessible concurrently by each of the processors. The search space is partitioned among the computing nodes, and sub-partitioned among the processors of each computing node. If a processor completes its sub-partition, it requests another sub-partition. The partitioning and sub-partitioning may be performed dynamically, and adjusted in real time.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 15/139,960, titled “FREQUENT PATTERN MINING” and filed Apr. 27, 2016, which is a continuation of U.S. patent application Ser. No. 13/095,415 (U.S. Pat. No. 9,348,852), titled “FREQUENT PATTERN MINING” and filed Apr. 27, 2011, the disclosures of which are hereby incorporated by reference in their entirety.

BACKGROUND

“Frequent patterns” are sequences of data items that occur in a database at a relatively high frequency. Data items may be numbers, characters, strings, names, records, and so forth.

Discovery of frequent patterns, also referred to as frequent pattern searching or mining, has become important in many fields, and it is often desired to find frequently occurring patterns in very large data sets.

One way to visualize the process of pattern mining is as a hierarchical graph or tree of patterns and sub-patterns. Suppose, for example, that it is desired to find frequently occurring character patterns in a text. A first pass might identify all single characters that might form the beginning of a character pattern. These “candidate” items would form the first level of a hierarchical tree structure. A second pass might then add a second, dependent level to the hierarchical tree structure. The second pass would find, for each first-level candidate item, all single characters or strings that occur subsequent to the first-level candidate item. This process would then be iterated to add further sub-levels to the hierarchical tree, which could eventually be examined to find those strings or patterns that occur most frequently.

Many algorithms are available for implementing the process of searching for frequently occurring patterns. However, frequent pattern mining against large databases is computationally expensive and time consuming. Accordingly, efforts have been made to utilize multiple computers or computing nodes, running in parallel, to speed the process. A traditional approach to distributing tasks among computing nodes might be to partition the search space into many sub-search spaces, and utilize available computing nodes to search the partitions in parallel. However, it can be difficult to predict the amount of work that will be involved in processing any particular partition, and it is therefore difficult to create partitions in such a way that each computing node will have the same amount of work. The resulting unbalanced partitioning tends to decrease the efficiency of parallel pattern mining algorithms.

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description is set forth with reference to the accompanying figures. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. The use of the same reference numbers in different figures indicates similar or identical items.

FIG. 1 is a block diagram of an example system for mining frequently occurring patterns.

FIG. 2 is a schematic diagram illustrating a frequent pattern search technique.

FIG. 3 is a flow diagram showing a procedures for partitioning and assigning a frequently occurring pattern search to multiple computing nodes and processors.

FIG. 4 is a flow diagram illustrating reallocation of search sub-partitions among processors of a computing node.

FIG. 5 is a flow diagram illustrating reallocation of search partitions among computing nodes.

SUMMARY

Frequent pattern mining is conducted using a two-layer architecture. A first level of tasks is distributed to a plurality of computing nodes: the search space is partitioned, and one or more of the resulting partitions are assigned to each of the computing nodes. Each computing node has a plurality of processors.

A second level of tasks is distributed to the processors within the computing nodes: the partition of the search space assigned to a particular computing node is sub-partitioned, and one or more sub-partitions are assigned to each of the processors of the computing node.

The frequent pattern mining is conducted against a data set that is stored in shared high-speed memory of each of the computing nodes, for concurrent or shared access by the processors of each computing node.

This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.

DETAILED DESCRIPTION

FIG. 1 shows an example of a computer system 100 configured to perform frequent pattern mining among items of a data set. The computer system 100 includes two levels or layers of computing entities. At a first level, a plurality of computing nodes 102 communicate with each other over a network 104 or other type of communication channel. At a second level, each of the computing nodes 102 has multiple processors that perform portions of the frequent pattern mining.

The lower portion of FIG. 1 shows an example configuration of a single computing node 102. Each of the computing nodes 102 has generally the same configuration.

Each computing node 102 may comprise a conventional computer having multiple processors or CPUs (central processing units) 106. For example, a single computing node may utilize 16 or more processors. Each computing node 102 may also have various types of memory, some of which may be used or allocated as shared memory 108 and as in-process memory 110.

The shared memory 108 and in-process memory 110 in many embodiments may comprise electronic and/or semiconductor memory such as volatile, randomly-addressable memory or RAM that is accessible locally to the computing node 102 by means of a local bus or communications channel (not shown). This type of memory is frequently referred to as the computer's “RAM,” and in many embodiments will be formed by high-speed, dynamically-refreshed semiconductor memory.

Each computing node 102 may also have access to other types of memory (not shown), including read-only memory (ROM), non-volatile memory such as hard disks, and external memory such as remotely located storage, which may provide access to various data, data sets, and databases. Various computing nodes 102 may also be capable of utilizing removable media.

In the described embodiment, the shared memory 108 is accessible concurrently by all of the processors 106, and contains a data set 112 which is to be the object of a frequently-occurring pattern search. The data set 112 may in some embodiments take the form of a structured database. For example, the data set 112 may comprise a SQL (structured query language) database or some other type of relational database that is accessible using conventional database query languages.

The data set 112 contains a plurality of data items, and each data item is formed by one or more elements. The individual data items may comprise text, strings, records, and so forth. Elements within data items may comprise characters, words, lines, names, etc. The object of frequent pattern mining is to identify patterns of elements that occur frequently in different items of the data set. For example, it may be desired to find the sequences of characters that occur most frequently in string items, or to find frequently occurring sequences of function names that occur in program execution logs.

The shared memory 108 may also contain pre-calculated, static data 114 related to or used by frequent pattern mining algorithms.

Both the data set 112 and the pre-calculated, static data 114 may be accessed by any of the processors 106.

Because of the decreasing cost and increasing densities of computer memory, the shared memory 108 may be quite large. In current embodiments, the combined shared memory 108 and in-process memory 110 may be 48 gigabytes or more, which is large enough to contain a very large data set without needing memory swapping or paging. Future technologies will undoubtedly increase the practical amounts of RAM available within single computing nodes.

While the shared memory 108 is accessible in common by the multiple processors 106, each instance of the in-process memory 110 is dedicated and private to an individual one of the processors 106 or to one or more of the processes being executed by the processors. The in-process memory 110 stores dynamic variables 116 and other data that may be generated and maintained by processes executed by the processors 106. Note that the in-process memory 110 may in some embodiments include paged memory.

The embodiment described herein utilizes task partitioning, so that frequent pattern mining can be partitioned and performed in parallel by different computing nodes 102 and processors 106. Using this approach, each processor 106 of a single computing node 102 has access to all records or data items of the data set, but is responsible for a different portion or partition of the search space.

Tasks are assigned in two stages. At a first stage, the work of a frequent pattern search is divided into multiple tasks, which are assigned to computing nodes. At a second stage, each of these tasks is divided into sub-tasks, which are assigned to individual processors of the computing nodes. The task division may be performed at a level of granularity that allows a number of tasks or sub-tasks to be reserved for future assignment as computing nodes or processors complete their current assignments.

Each task involves searching for frequent patterns in a partition or sub-partition of the overall search space. Partitioning and sub-partitioning are performed with an effort to produce partitions and sub-partitions of equal size, so that computing nodes and processors are assigned equal amounts of work. To account for estimation inaccuracies, initial partitions and sub-partitions can be made sufficiently small so that some partitions and sub-partitions are held in reserve, for future assignment. When a computing node or processor completes its current assignment, it may request a further assignment. This request may be satisfied by the assignment of an as-yet unassigned partition or sub-partition, if available. If no unassigned partitions or sub-partitions are available, the system may re-partition or sub-partition an existing assignment, and may reassign one of the resulting partitions or sub-partitions to a requesting computing node or processor.

The searching itself can be performed in different ways, using various algorithms. For example, certain embodiments may utilize the frequent pattern mining algorithm described in the following published reference:

-   -   Jianyong Wang and Jiawei Han. 2004. BIDE: Efficient Mining of         Frequent Closed Sequences. In Proceedings of the 20th         International Conference on Data Engineering (ICDE '04). IEEE         Computer Society, Washington, D.C., USA, 79-.         Other algorithms might also be used.

A frequent pattern mining algorithm such as this involves building a hierarchical pattern tree by exploration, starting with high levels and building through lower and yet lower levels.

FIG. 2 illustrates an initial or early definition of a pattern search space 200. The search space begins at an empty root level 202. Exploration of data items (which in this example are strings) reveals a first level 204 of the search space, with nodes corresponding to characters that may form the first elements of frequently occurring element patterns: “A”, “B”, and “C”. Further exploration of the data items identifies a second level 206 of the search space, having nodes corresponding to characters that may follow the initial characters of the first level 204. For example, the characters “Z” and “F” have been found in the data set to follow occurrences of “A”. The second level can alternatively be viewed as having nodes that correspond to sub-patterns, where the sub-pattern corresponding to a particular node is a concatenation of the elements of those nodes found along the path from the root node to the particular node. For example, a first-level node may corresponds to the pattern “A”, and the second level nodes below the first-level node “A” might correspond to sub-patterns “AZ” and “AF”, respectively.

Dashed lines leading from the nodes of the second level 206 indicate the possible existence of yet lower-level nodes and sub-patterns, which are as yet unexplored and thus unknown.

A node having dependent nodes can be referred to as a parent node. Nodes that depend from such a parent node can be referred to as child nodes or children. A node is said to have “support” that is equal to the number of data items that contain the sub-pattern defined by the node. In many situations, “frequently” occurring patterns are defined as those patterns having support that meets or exceeds a given threshold.

Given a search space definition as shown in FIG. 2, further exploration can be partitioned into separate tasks corresponding to the nodes of one of the levels of the defined space 200. For example, further exploration can be separated into three tasks corresponding to the three first-level nodes “A”, “B”, and “C”. Each task is responsible for finding sub-patterns of its node. Alternatively, the further exploration might be partitioned into six tasks, corresponding to the nodes of the second level 206 of the search space. This type of partitioning can be performed at any level of the search space, assuming that exploration has been performed to reveal that level of the search space.

Referring again to FIG. 1, each of the processors 106 may be configured to execute a frequent pattern searching algorithm in a search task or process 118. In the described embodiment, the search space is partitioned as described above, and partitions of the frequent pattern search are assigned to each of the computing nodes 102. Sub-partitions of these partitions are then defined, based on lower-level nodes of the search space, and are assigned as tasks to each of the processors 106. Each processor conducts its sub-partition of the search against the data set 112, which is stored in the shared memory 108.

Note that in this embodiment, the entire data set 112 (containing all data items) is replicated in the shared memory 108 of each computing node 102, so that each search task 118 has access to the entire data set.

The computing nodes 102 include a head node 120 that executes a scheduler 122 to allocate partitions of the frequent pattern search to individual computing nodes 102. In addition, the processors 106 of each computing node 102 include a head processor 124 that executes a scheduler 126 to allocate sub-partitions of the frequent pattern search to individual processors 106 of the computing node 102. The head node 120 and the head processors 124 also dynamically reallocate the portions and sub-portions of the pattern search upon demand. Reallocation takes place first among the processors 106 of individual computing nodes 102, and secondarily among the computing nodes 102 when reallocation within a computing node is undesirable or impractical.

FIG. 3 illustrates an initial assignment or allocation 300 of tasks to computing nodes 102 and their processors 106. An action 302 comprises partitioning the overall search space into a plurality of partitions. This is performed as described above, by exploring and growing the search space to a predetermined level of granularity. In most cases, relatively high level nodes will be used to define the initial partitions of action 302.

At 304, the head node 120 assigns one or more of the initial partitions to each of the computing nodes 102. All identified partitions may be assigned at this point, or some partitions may be reserved for future assignment when individual computing nodes complete their initial assignments.

At 306, the head processor 124 of each computing node 102 sub-partitions any partitions that have been assigned to it, creating multiple sub-partitions. The head processor 124 uses techniques similar to those used by the head computing node 120 to identify sub-partitions, by exploring and growing the search space to identify sub-nodes or next-lower level nodes—nodes at a level or levels below the search space levels that were used by the head computing node 120 to identify the initial partitions. At 308, the sub-partitions are assigned to individual processors 106 of the computing nodes, by the head processor 124 of each computing node. All of the identified sub-partitions may be assigned at this point, or some sub-partitions may be reserved for future assignment when individual processors complete their initial assignments.

FIG. 4 illustrates an example process 400 for dynamically reallocating sub-partitions to individual processors 106. This process is initiated when a processor completes its current assignment, and thus runs out of work to perform. These actions are performed by the scheduler 126 of an individual computing node 102. The processor 106 that has run out of work will be referred to as a free processor. Other processors within the computing node will be referred to as busy processors.

At 402, the scheduler 126 determines whether any sub-partitions remain unassigned, resulting from any previous sub-partitioning efforts. If so, an action 404 is performed, comprising assigning one of these available sub-partitions to the free processor. The free processor commences searching in accordance with the assignment.

If there are no remaining unassigned sub-partitions, the scheduler determines at 406 whether it is desirable for one of the busy processors to relinquish part if its previously allocated sub-partition. This can accomplished by querying each of the busy processors to determine their estimated remaining work. Whether or not it is desirable to further sub-partition the work currently being processed by a busy processor is evaluated primarily based on the estimated work remaining to the busy processor. At some point, a processor will have so little work remaining that it will be inefficient to further sub-partition that work.

If at 406 there is at least one busy processor with sufficient remaining work that it would be efficient to sub-partition that remaining work, execution proceeds with the actions shown along the left side of FIG. 4. An action 408 comprises selecting one of the busy processors 106. This may be accomplished by evaluating the work remaining to each of the processors, and selecting the processor with the most remaining work.

At 410, the scheduler 126 or the selected busy processor itself may sub-partition the remaining work of the busy processor. For example, the remaining work may be sub-partitioned into two sub-partitions, based on currently known levels of the search space that the busy processor is currently exploring. At 412, one of the new sub-partitions is assigned to the free processor.

If at 406 there is not at least one busy processor with sufficient remaining work that it would be efficient to sub-partition that remaining work, execution proceeds with the actions shown along the right side of FIG. 4. An action 414 comprises requesting a new partition assignment or reassignment from the scheduler 122 of the head node 120. An action 416 comprises sub-partitioning the new assignment, using the techniques already described. An action 418 comprises assigning one of the resulting sub-partitions to the free processor. The remaining sub-partitions are held by the scheduler 126 for future assignment to other processors as they complete their current assignments.

FIG. 5 illustrates an example process 500 for dynamically reallocating search space partitions to individual computing nodes 102. This process is initiated upon receiving a request from a computing node, such as indicated at 414 of FIG. 4. These actions are performed by the scheduler 122 of the head node 120. The requesting computing node 102 that has run out of work will be referred to as a requesting computing node. Other computing node will be referred to as busy computer nodes.

At 502, the scheduler 122 determines whether any partitions remain unassigned, resulting from any previous partitioning efforts. If so, an action 504 is performed, comprising assigning one of these available partitions to the free computing node. The free computing node commences searching in accordance with the assignment, as described with reference to FIG. 4.

If there are no remaining unassigned partitions, the scheduler determines at 506 whether it is desirable for one of the busy computing nodes to relinquish part if its previously allocated partition. This can accomplished by querying each of the busy computing nodes to determine their estimated remaining work. Whether or not it is desirable to further partition the work currently being processed by a busy computing node is evaluated primarily based on the estimated work remaining to the busy computing node. At some point, a computing node will have so little work remaining that it will be inefficient to further partition that work. Note also that reassigning work from one computing node to another involves the busy computing node reassigning or redistributing work to among its individual processors.

If at 506 there is not at least one busy computing node with sufficient remaining work that it would be efficient to partition that remaining work, an action 508 is performed of simply waiting for the remaining computing nodes to complete their work. Otherwise, execution proceeds with the actions shown along the left side of FIG. 5. An action 510 comprises selecting one of the busy computing nodes 102. This may be accomplished by evaluating the work remaining to each of the computing nodes, and selecting the computing node with the most remaining work.

At 512, the scheduler 122 or the selected busy computing node itself may partition the remaining work of the busy computing node. For example, the remaining work may be partitioned into two sub-partitions, based on currently known sub-levels of the search space that the busy processor is currently exploring. At 514, one of the sub-partitions is assigned to the free computing node.

Using the techniques described above, reassignment of partitions and sub-partitions is performed dynamically, and is initiated when a processor or computing node completes its current assignment.

Partitioning, assignment, and reassignment may involve evaluating the amount of work associated with individual partitions or sub-partitions—also referred to as the “size” of the partition or sub-partition. In practice, the actual size of any partition is unknown, because that partition has not yet been fully explored, and only a complete exploration will reveal the size. However, partition and sub-partition sizes can be estimated or predicted.

More specifically, each partition or sub-partition may correspond to a sub-pattern of the search space. The support of the sub-pattern—the number of data items that contain the sub-pattern—is used on some embodiments as an estimate of the size of the partition. Partitions with higher support are predicted be larger than partitions with lower support. Alternatively, the sum of supports of the next-lower level nodes of the search space may be used to estimate the size of the sub-pattern. As a further alternative, for example when the algorithm in the reference cited above is used, the average sequence length of the projection database of immediate next-lower level nodes of the search space may be used as an indication or estimate of partition size.

Other types of estimations may be used in other embodiments.

Generally, reallocations and reassignments should be performed according to criteria that account for efficiency. For example, reassignments among the processors of computing nodes should be performed at a higher priority than reassignments among computing nodes. Furthermore, any reassignments should be performed in a way that contributes to balanced workloads among the processors and computing nodes. Also, granularity of reassignments should not be too small, because each reassignment involves significant overhead.

In some embodiments, the schedulers 122 and 126 may monitor remaining workload of the various computing nodes and processors. When work is reallocated, the schedulers account for this in their estimations. Furthermore, the schedulers may maintain estimation models to predict the remaining work of individual computing nodes and processors. The estimation models may be updated or adjusted in response to actual performance of the searching, so that the models become more accurate over time.

Illustrative systems and methods of conducting frequent pattern mining are described above. The illustrative methods are illustrated as a collections of blocks in logical flow graphs, representing sequences of operations that can be implemented in hardware, software, firmware, or a combination thereof. Certain features of the systems and acts of the methods need not be arranged in the order described, may be modified, and/or may be omitted entirely, depending on the circumstances.

Also, any of the acts described above with respect to any method may be implemented by a processor or other computing device based on instructions stored on one or more computer-readable storage media. Computer-readable media includes at least two types of computer-readable media, namely computer storage media and communications media.

Computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules, or other data. Computer storage media includes, but is not limited to, phase change memory (PRAM), static random-access memory (SRAM), dynamic random-access memory (DRAM), other types of random-access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technology, compact disk read-only memory (CD-ROM), digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transmission medium that can be used to store information for access by a computing device.

In contrast, communication media may embody computer readable instructions, data structures, program modules, or other data in a modulated data signal, such as a carrier wave, or other transmission mechanism. As defined herein, computer storage media does not include communication media.

Furthermore, although the disclosure uses language specific to structural features and/or methodological acts, the invention is not limited to the specific features or acts described. Rather, the specific features and acts are disclosed as illustrative forms of implementing the invention. 

The invention claimed is:
 1. A system comprising: a first computing node configured to allocate a first plurality of partitions corresponding to a pattern search space, each partition of the first plurality of partitions corresponding to a portion of the pattern search space; and a plurality of second computing nodes configured to receive and process respective partitions of the first plurality of partitions allocated by the first computing node, wherein each of the plurality of second computing nodes comprise: a machine-readable memory configured to store at least one partition allocated from the first plurality of partitions by the first computing node; a first processor configured to: determine a second plurality of partitions from the at least one partition; and allocate the second plurality of partitions, wherein the second plurality of partitions correspond to a portion of the pattern search space; a plurality of processors configured to process the second plurality of partitions allocated by the first processor; and wherein the first processor is further configured to: query the plurality of processors to determine a remaining time for each processor of the plurality of processors to process its allocated partition; and re-allocate a portion of a partition allocated to a processor of the plurality processors based on the remaining time for the processor to process the allocated partition.
 2. The system of claim 1, wherein the first processor is further configured to: receive an instruction from a processor selected from the plurality of processors that the processor has completed its allocated partition; determine whether there a remaining partitions selected from the second plurality of partitions; and allocate a partition selected from the second plurality of partitions in response to a determination that there are remaining partitions.
 3. The system of claim 1, wherein the first computing node is configured to determine the first plurality of partitions to a first predetermined level of granularity that corresponds to a first node level of the pattern search space.
 4. The system of claim 3, wherein: the first processor is configured to determine the second plurality of partitions to a second level of granularity that corresponds to a second node level of the pattern search space; and the second level of granularity is more granular than the first level of granularity.
 5. The system of claim 1, wherein the first processor is further configured to reserve a third plurality of partitions selected from the second plurality of partitions, wherein the third plurality of partitions are allocated after each of the processors of the plurality of processors have been allocated a partition selected from the second plurality of partitions.
 6. The system of claim 1, wherein the first processor is configured to dynamically re-allocate the second plurality of partitions in response to a notification that at least one processor of the plurality of processors has completed its allocated partition.
 7. The system of claim 1, wherein the first computing node is configured to dynamically re-allocate the second plurality of partitions to one or more computing nodes selected from the plurality of computing nodes in response to a notification that one of the plurality of second computing nodes is unable to complete one or more partitions selected from the second plurality of partitions.
 8. The system of claim 1, wherein the portion of the partition that is re-allocated is selected from a processor having the most remaining time to process its allocated partition relative to the remaining time of the other processors of the plurality of processors having an allocated partition.
 9. The system of claim 1, wherein the first computing node is further configured to: query the plurality of second computing nodes to determine a remaining time for each computing node to process its allocated partition, and re-allocate a portion of a partition allocated to a computing node of the plurality of computing nodes based on the remaining time for the computing node to process the allocated partition.
 10. A method comprising: allocating, by a first computing node, a first plurality of partitions to a plurality of computing nodes, wherein each partition of the first plurality of partitions corresponding to a portion of a pattern search space; for each of a plurality of second computing nodes: storing, by a machine-readable memory, at least one partition allocated from the first plurality of partitions; determining, by a first processor of the second computing node, a second plurality of partitions from the at least one partition; allocating, by the first processor, the second plurality of partitions to a plurality of processors of the second computing node, wherein the second plurality of partitions correspond to a portion of the pattern search space; and processing, by the plurality of processors, the second plurality of partitions allocated by the first processor; and wherein the first processor is further configured to: query the plurality of processors to determine a remaining time for each processor of the plurality of processors to process its allocated partition; and re-allocate a portion of a partition allocated to a processor of the plurality processors based on the remaining time for the processor to process the allocated partition.
 11. The method of claim 10, further comprising: receiving, by the first processor, an instruction from a processor selected from the plurality of processors that the processor has completed its allocated partition; determining, by the first processor, whether there a remaining partitions selected from the second plurality of partitions; and allocating, by the first processor, a partition selected from the second plurality of partitions in response to a determination that there are remaining partitions.
 12. The method of claim 10, further comprising: determining, by the first computing node, the first plurality of partitions to a first predetermined level of granularity that corresponds to a first node level of the pattern search space.
 13. The method of claim 12, further comprising: determining, by the first processor, the second plurality of partitions to a second level of granularity that corresponds to a second node level of the pattern search space, wherein the second level of granularity is more granular than the first level of granularity.
 14. The method of claim 10, further comprising: reserving, by the first processor, a third plurality of partitions selected from the second plurality of partitions, wherein the third plurality of partitions are allocated after each of the processors of the plurality of processors have been allocated a partition selected from the second plurality of partitions.
 15. The method of claim 10, further comprising: dynamically re-allocating, by the first processor, the second plurality of partitions in response to a notification that at least one processor of the plurality of processors has completed its allocated partition.
 16. The method of claim 10, further comprising: dynamically re-allocating, by the first computing node, the second plurality of partitions to one or more computing nodes selected from the plurality of second computing nodes in response to a notification that the second computing node is unable to complete one or more partitions selected from the second plurality of partitions.
 17. The method of claim 10, wherein the portion of the partition that is re-allocated is selected from a processor having the most remaining time to process its allocated partition relative to the remaining time of the other processors of the plurality of processors having an allocated partition.
 18. The method of claim 10, further comprising: querying, by the first computing node, the plurality of second computing nodes to determine a remaining time for each computing node to process its allocated partition; and re-allocating, by the first computing node, a portion of a partition allocated to a computing node of the plurality of second computing nodes based on the remaining time for the computing node to process the allocated partition. 